A “pure” Möbius strip would have no thickness, but I prefer to deal with physical reality rather than mathematical concepts. If you want an exhaustive (and exhausting) mathematical treatment of the Möbius strip, I recommend

A simple Möbius strip is easily made using a strip of paper with a half-twist, then connect it’s ends together, and I use these as models for my woodcarving projects.

The Möbius strip has two basic forms:
[] A right-hand twist, and . . .
[] A left-hand twist.

There are virtually endless variations possible with the Möbius strip. They can be skinny or fat, long or short, single twist or multiple twists, and can be made to resemble various objects; thus providing more carving opportunities than one can ever achieve. For example, here is one that I call a “Möbius Bird”.

In Möbius Strips, Part 2 we (YOU and I ) will take a look at a few cravings of Möbius strips, both finished works, and one that is a work in progress.

The final four cuts are similar to the previous ones in that the layout lines simply run from the centerline of the circle on one surface to the tangent point on the circle on the adjacent surface. as shown in the photo, above.

As you can see in the photo, the cut line must ” bend ” around a corner on its way from the center line on the upper surface, then go to the point at which the tangent line met the circumference of the circle on the adjacent surface.

Unfortunately, the circle no longer exists on two of the edges (the vertical edges in the photo) because it was cut away when previous cuts were made.

Fortunately, the length of the edges of each of the twelve pentagons (the length of the diameter of the layout circle on the “master” surface of the cube you started with) was captured in Part 1 of this series. Aren’t you glad you captured that length now that it is useful for finding exactly where the cut line must terminate?

Simply divide the diameter in half to find the midpoint, then place the midpoint of the diameter at the midpoint of the long edge on the right hand side of the photo, then mark the ends of the diameter. This locates the tangent points for the new layout lines.

Distortion in the photo makes it appear that the edges of the pentagons are not equal, but they are.

I use a very thin and flexible strip of teflon plastic straightedge to serve as a guide which bends around the ‘corner’ to connect the two points, then draw the layout line from the end of the center line on top of the block to the tangent points.

If you don’t have a suitable straightedge for bending around the “corner”, here is an alternate method – –

Use that circle’s diameter (whatever it is for YOUR dodecahedron) then:

[] place one end at the end of the center line on top of the block

[] slide the other end along the edge until it just reached the point where the ” bend” must be made.

[] draw a layout line from the end of the center line at the top of the block to the “bend” point, then . . .

[] draw another layout line from the “bend” point to the tangent point on the “long edge”

Repeat the steps outlined above for the remaining layout lines, and you are ready to do the final cuts.

BUT . . . before you apply saw to wood, I suggest you check the lengths of the edges for the pentagons to be sure they are identical to the diameter of the layout circle you first drew in Part 1.

Make any correction(s) required before you actually do the cuts.

After doing the final four cuts, you will probably have a bit of touch-up to do before applying varnish or whatever you decide to use for the finished surface. I usually apply three or more coats of high quality marine grade spar varnish, followed by a couple coats of paste floor wax as the finish for my wooden dodecahedrons.

Speaking of a finished dodecahedron, here is a photo of one that I finished recently, then attached atop a posts supporting the railing around the deck outside one of our patio doors . . .

. . . and here’s a couple of smaller dodecas that have been jazzed up a bit with contrasting colored dots in the middle of each pentagonal face . . .

The photo above shows the result of making cuts 1, 2, 3, and 4 on the way to making a dodecahedron from a wooden cube, as detailed in Part 1.

The layout lines for cuts 5, 6, 7, and 8 will look something like the photo shown below.

I drew the lines shown here on a photo using the mouse on my computer because the actual lines did not show up well.

As you can see, the cut line across the top of the block follows the tangent lines you drew on the ” master ” face of the cube, then down across the cut surface to the diameter line of the circle on the adjacent face, then across the circle to the far cut surface.

An identical set of layout lines are on the opposite side of the block.

After making cuts 5, 6, 7, and 8 your block will look something like this . . .

The photo, above, shows a newly cut dodecahedron, fresh from the workbench, ready for sanding, varnishing, waxing, and polishing. In other words, it is about half way to being ready for prime time display.

Making a wooden dodecahedron is not a difficult task, but it is time-consuming, and requires careful attention to detail. I have never kept a detailed log of exactly how much time is required from start to finish, but I suspect it is at least a dozen hours, or so, if all the time for various tasks is included; tasks such as selecting wood, layout, cutting, sanding, varnishing, etc.

First, a quick review for those whose high school geometry has become a bit rusty (scroll down to ‘ Begin With a Cube ‘, below, if you don’t need a review).

There are five Platonic solids:

The Hexahedron, AKA Cube (three squares at each vertex)
The Tetrahedron (three triangles at each vertex)
The Octahedron (four triangles at each vertex)
The Dodecahedron (three pentagons at each vertex)
The Icosahedron (five triangles at each vertex)

Notice that the dodecahedron is the only Platonic solid that has pentagonal faces.

A dodecahedron has twelve faces that are congruent regular pentagons with three meeting at each
vertex. Regular means that the sides of the pentagons are all the same length, and Congruent means that the are all the same size.

The ancient Greeks recognized that there are only five platonic solids. But why are there only five?

The key observation is that the interior angles meeting at each vertex of the Platonic solids add to less than 360 degrees.

Hexahedron: Three squares at each vertex; 3×90 = 270

Tetrahedron: Three triangles at each vertex; 3×60 = 180 degrees

Octahedron: Four triangles at each vertex; 4×60 = 240 degrees

Dodecahedron: Three pentagons at each vertex; 3×108 = 324 degrees

Icosahedron: Five triangles at each vertex; 5×60 = 300 degrees

If you feel you need or want more detail regarding Platonic solids in general and dodecahedrons in particular, there are many fine websites that can provide more than you ever wanted to know. I think wikipedia is a good place to start, or simply Google the name, dodecahedron.

As with most any artifact, there are several ways to go about making a dodecahedron. For me, the most satisfying method is to cut a dodecahedron from a block of wood.

BEGIN WITH A CUBE

The ” block of wood ” is one of the platonic solids: the hexahedron, commonly called a cube.

There are a number of ways to go about trimming away the ” excess” wood from the cube to reveal the dodecahedron within a cube. One method I have used successfully to build several dodecahedrons is detailed below.

The more accurate your cube, the more accurate your dodecahedron will be. Be sure your cube is square (all angles at the 8 corners are 90 degrees ). This will result in all the edges being exactly the same length. It is relatively unimportant what, exactly, the ” same length ” is.

The cube shown above measures about 8.2 cm on each edge, and was prepared from a ” 4×4 ” that was used as a post in an old redwood fence that I replaced several months ago.

I make most of my dodecahedrons from salvaged lumber, but any lumber yard worthy of the name can provide suitable lumber. One might expect new, store-bought lumber to be of uniform dimensions, but I have found this is not so. In any event, you will need to trim your cube to make it as close to perfect as possible. Stock lumber in the form of a 4×4 or 6×6 or 8×8 is a good place to start. If money is no object, excellent wood for carving can be had from specialty suppliers.

[] Divide each of the six faces of the cube into four squares.

[] Choose any one of the six square faces of your cube to be the ‘ master ‘ layout surface.

[] Where the layout lines cross is the midpoint of this surface, point ‘ m ‘.

You should use fine lines in order to increase the accuracy of your layout. I have found that a mechanical pencil using 0.5 mm lead works well for this purpose. Tiny errors accumulate during the twelve cuts you will be making, and this will cause your finished dodecahedron to be out of whack.

Take your time and get the layout lines as close to perfect as possible because these layout lines will by your guide for the twelve cuts required to turn a cube into a dodecahedron.

[] Divide one-half of the Horizontal axis in half. This is point ‘ h ‘.

[] The top of the vertical axis is point ‘ t ‘.

[] Place the point of your compass on point ‘ h ‘ and the pencil on point ‘ t ‘, then draw an arc that intercepts the other half of the segment at point ‘ i ‘.

[] Use your compass to capture the distance from point ‘ i ‘ to the edge of the square at point ‘e ‘.
If your compass has a locking mechanism, lock your compass now to capture the length, i – e.
Call this captured length ‘ r ‘, for radius.

NOTE: Points h, t, i, and e have now served their purpose, so you can forget about them – – after first making sure you have captures the distance i – e for future reference.

Another NOTE: You will find the length of both the radius and the diameter of the circles to be essential for additional layout lines and for checking the accuracy of your cuts. More about this, later.

[] Place the point of your compass at ‘ m ‘, and draw a circle with a radius of ‘ r ‘.

Congratulations!

The diameter of this circle is the length of the edges of each of the twelve pentagonal surfaces on your dodecahedron. You can use this fact to check the perfection of your dodecahedron when you are finished, but don’t be concerned about that at this time. There are still many steps to complete before checking the accuracy of your finished work.

[] Now, draw a circle on each of the remaining sides of your cube. TAKE CARE to insure the compass point is set EXACTLY where the horizontal and vertical lines cross. Tiny variations here will affect the following layout and cuts.

If your compass is not lockable, check the setting against the ‘ master ‘ surface before drawing each circle to insure that the radius is the same on all surfaces of the cube.

Remember to keep your layout lines as narrow as possible because they will determine the accuracy of your cuts.

[] The edges on opposite sides of the cube will be parallel to each other.

[] The edges on adjacent surfaces of the cube will be perpendicular to each other, as illustrated in the drawing, below.

Layout lines, tangent to the circles, can now be marked on each of the six surfaces of the cube.

Alert readers, such as yourself, will recognize that parts of these lines will be cut away. That is true

-BUT-

. . . the parts that are not cut away will be useful for laying out further cuts.
THE FIRST FOUR CUTS

OK, now we (YOU and I) can draw lines for our first four cuts. NOTICE that the lines run from the center line if the circle on one surface to the circumference of the circle on the adjacent surface.

The shaded areas will be cut away from the cube.

My favorite tool for making the cuts is a Japanese style handsaw that cuts on the back- stroke. I strongly recommend that an electric-powered miter saw NOT be used to make the cuts, because it is far too dangerous when working with such small, blocky pieces.

Be that as it may, here are a couple of photos showing what your former cube will look like after the first four cuts have been made and trimmed to match the layout lines.

In Part 2 of Making a Dodecahedron, we will layout and do cuts five, six, seven, and eight.